Fractional nuclear charge approach to isolated anion densities for Hirshfeld partitioning methods

Abstract

Atoms in molecules methods that rely on reference promolecular densities typically require that one define, or otherwise determine, the densities of unbound atomic anions. Whereas the isolated atomic polyanions are always physically and computationally unbound, monoanions can be either physically bound but computationally unbound (like the oxygen anion at the Hartree-Fock level of theory), or physically unbound but computationally bound (like the nitrogen anion using many DFT methods with a basis set including diffuse functions). Depending on the level of theory and basis set used, the densities of negatively charged atomic ions can decay very slowly and even be nonmonotonically decreasing. These delocalized anionic densities induce ill-behaved atomic properties for compounds containing highly reduced atoms. To treat the problem of unphysical proatom densities in iterative Hirshfeld methods, we compute the smallest (typically fractional) nuclear charge to bind all electrons, called the effective nuclear charge ${Z}_{\mathrm{A}}^{\mathrm{eff}}$ of an atom A. When ${Z}_{\mathrm{A}}^{\mathrm{eff}}>{Z}_{\mathrm{A}}$ at a given level of theory, the scaled density corresponding to the effective nuclear charge is used as the negatively charged proatom density. This novel approach dramatically improves the computational robustness of the iterative Hirshfeld partitioning scheme.

Appendix

Here we demonstrate how density is scaled to have the right cusp according to Eq. (10). The exact cusp relation for the N _A-electron ground-state electron density of an atom with charge ${Z}_{\mathrm{A}}^{\mathrm{eff}}$is

$$ -2{Z}_{\mathrm{A}}^{\mathrm{eff}}={\left[\frac{1}{\rho_{\mathrm{A}}\left(r,\theta, \phi; {N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right)}\frac{\partial {\rho}_{\mathrm{A}}\left(r,\theta, \phi; {N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right)}{\partial r}\right]}_{r=0} $$

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Scaling the density according to Eq. (10) fixes the cusp condition,

$$ {\displaystyle \begin{array}{l}{\left[\frac{1}{{\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}\right)}^3{\rho}_{\mathrm{A}}\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}r,\theta, \phi, {N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right)}\frac{\partial {\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}\right)}^3{\rho}_{\mathrm{A}}\left(\frac{Z_A}{Z_A^{\mathrm{eff}}}r,\theta, \phi, {N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right)}{\partial r}\right]}_{r=0}\\ {}=-2{Z}_{\mathrm{A}}^{\mathrm{eff}}\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}\right)\\ {}=-2{Z}_{\mathrm{A}}\end{array}} $$

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while maintaining the normalization of the electron density, i.e.,

$$ {\displaystyle \begin{array}{l}{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}\right)}^3{\rho}_{\mathrm{A}}\left(\frac{Z_A}{Z_A^{\mathrm{eff}}}x,\frac{Z_A}{Z_A^{\mathrm{eff}}}y,\frac{Z_A}{Z_A^{\mathrm{eff}}}z,{N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right) dxdydz\\ {}={\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\rho}_{\mathrm{A}}\left(\frac{Z_A}{Z_A^{\mathrm{eff}}}x,\frac{Z_A}{Z_A^{\mathrm{eff}}}y,\frac{Z_A}{Z_A^{\mathrm{eff}}}z,{N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right)d\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}x\right)d\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}y\right)d\left(\frac{Z_{\mathrm{A}}}{Z_{\mathrm{A}}^{\mathrm{eff}}}z\right).\\ {}={\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }{\rho}_{\mathrm{A}}\left(X,Y,Z,{N}_{\mathrm{A}};{Z}_{\mathrm{A}}^{\mathrm{eff}}\right) dXdYdZ\\ {}={N}_A\end{array}} $$

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